Fracterm Calculus for Partial Meadows

Abstract

Partial algebras and datatypes are discussed with the use of signatures that allow partial functions, and a three-valued short-circuit (sequential) first order logic with a Tarski semantics. The propositional part of this logic is also known as McCarthy calculus and has been studied extensively. Axioms for the fracterm calculus of partial meadows are given. The case is made that in this way a rather natural formalisation of fields with division operator is obtained. It is noticed that the logic thus obtained cannot express that division by zero must be undefined. An interpretation of the three-valued sequential logic into $\perp$-enlargements of partial algebras is given, for which it is concluded that the consequence relation of the former logic is semi-computable, and that the $\perp$-enlargement of a partial meadow is a common meadow.

Comments: 27 pages, 10 tables. Main differences with v1: (p.4) reference [8, App.A.3 & A.4] to short proofs of DNE and (a)-(e) has been added; (p.9) the quoted theorem has been moved here and is followed by a comment; (p.17) Prop.3.3.4 and its proof are now formulated more simply and preceded by an explanation; (p.18) ψ_true(T) and ψ_true(F) are now defined, as are their ψ_false values